Materials analysis using X-rays provides accurate data in a number of applications and industries. X-ray fluorescence measurements allow the determination of the elemental composition of a sample. In some applications however this is not enough and there is a need not merely to determine the elemental composition but also to determine structure parameters such as the crystalline phases of a sample and X-ray diffraction is used in these cases. Typically, high resolution X-ray diffraction measurements are carried out in reflection mode, where an incoming beam of X-rays is incident on a first surface of a sample and the X-rays diffracted by a diffraction angle 2θ from the same surface of the sample are detected by a detector.
In some applications it is useful to be able take X-ray diffraction measurements in a transmission mode, where the X-rays are incident on a first surface of a sample and diffracted by a diffraction angle 2θ are measured after passing through the sample from the first surface to the opposite second surface.
A problem with making measurements in this transmission geometry is that the sample itself may be absorbing for X-rays. Therefore, it is difficult to carry out accurate quantitative analysis of the diffracted X-rays to determine the amount of any given phase of the sample, since the absorption of X-rays in the sample is not in general known. Small changes in the concentration of various components in the sample can cause significant changes in absorption. This is a problem for quantitative X-ray analysis designed to measure the quantity of a given component in the sample, since the amount of that component is unknown but will affect the absorption.
A yet further problem when measuring pressed powder samples is that the thickness d and density ρ of the pressed pellet, or their more commonly used product ξ=ρd (known as “mass thickness” or “surface density”) is not generally exactly known. A value of the mass thickness can be directly obtained as the ratio of the weight of the pellet to its surface. However, the resulting number is not accurate enough and will introduce large errors in the quantification of the transmission measurements.
Furthermore, in an industrial environment, it will be desired to make a pressed powder sample and then measure it as soon as possible. It is generally undesirable to have to make accurate measurements of mass thickness before carrying out X-ray measurement.
These considerations may be seen with reference to FIG. 1 which illustrates the theoretically calculated diffraction intensity for free lime as a function of sample thickness for three samples of standard cement clinker materials (Portland cement clinker) mixed with a wax binder for various binder percentages of 0%, 10%, 20% and 30%. Note that in spite of the fact that the samples of higher thickness contain more diffracting material—a sample of twice the thickness has twice the amount of free lime—the diffracted intensity is in fact less.
Realistic sample thicknesses (>3 mm) and dilution ratios (10%-20%) that guarantee robust samples in industrial applications are in the highly non-linear regime. This means that a small thickness deviation will have a large effect on the measured or calculated intensities. This is mainly the reason that an estimate of the mass thickness will produce poor results.
Further, as illustrated in FIG. 2, the diffraction intensity is also dependent on the exact composition. FIG. 2 shows three graphs for three different samples each of Portland cement clinker. In spite of the general similarity between the samples, the diffraction intensity still varies from sample to sample illustrating that the effect of absorption is a function of the exact composition which varies from sample to sample. At a thickness of about 3 mm a difference of about 8% in diffraction intensity is seen. This too makes calculating a quantitative measure of free lime concentration from diffraction measurements difficult.
The effects of a variable composition on quantitative measurement is known as matrix correction since it depends on the composition of the measured sample. It is in general difficult to calculate the matrix correction without any information on thickness and composition. There is therefore a need for a measurement method which avoids this difficulty.
Absorption of electromagnetic waves that pass directly through a medium without diffraction may be characterised by the Beer-Lambert lawI=I0e−μρd   (1)Where I0 is the original intensity, I the intensity after passing through the material, μ the mass attenuation coefficient of the material, ρ the material density and d the material thickness (i.e. the ray path length in the material).
Using the simple Beer-Lambert law it is possible to derive the effect of the absorption on the measured X-ray intensity simply by a single value, the value of the product μρd. However, in order to correct measured diffraction intensities for thickness and matrix variations both the product μρd as well as the mass absorption coefficient μ are required. Two samples with the same product μρd but (for example) different μ would appear to have the same attenuation according to a simple Beer-Lambert law but in this application do not give the same intensity of a measured diffraction.
There is therefore a need for a method that quantitatively carries out a thickness and matrix correction for X-ray diffraction.